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Conclusion

Alan Weir

in Truth Through Proof: A Formalist Foundation for Mathematics

This chapter summaries the key thesis of the book: that, given the Sense/Circumstance/World framework in the philosophy of language, we can show that mathematics expresses objective truths, but not ... More

This chapter summaries the key thesis of the book: that, given the Sense/Circumstance/World framework in the philosophy of language, we can show that mathematics expresses objective truths, but not by dint of delineating a mind-independent reality. Neo-formalism is held to incorporate sound aspects of rival views, such as classic formalism; neo-logicism, in the idea that stipulation of axioms grounds mathematical truth; constructivism, insofar as the constructivist links truth with provability; strict finitism to the extent that the strict finitist emphasizes that the ontology of mathematics can include only a finite corpus of concrete tokens. Neo-formalism even incorporates elements of platonism, insofar as it upholds the objectivity of mathematical truth. However neo-formalism, in rejecting the platonistic realm of an external mathematical ontology as a mythological projection of human activity, avoids the crippling metaphysical and epistemological problems of platonism.Less

Proof Set in Concrete

Alan Weir

in Truth Through Proof: A Formalist Foundation for Mathematics

This chapter attempts to carry out the task promised earlier: a purely concretist syntax and proof theory, where abstract types are replaced by concrete ‘tipes’, construed as fusions of concrete ... More

This chapter attempts to carry out the task promised earlier: a purely concretist syntax and proof theory, where abstract types are replaced by concrete ‘tipes’, construed as fusions of concrete tokens. The conclusion is drawn that there are only finitely many sentences and proofs in actual human languages, and since no appeal to possibility beyond everyday uncontentious uses is made this seems to drive us towards a strict finitism. In particular, how can those concrete utterances which, given limits on the size of graspable proofs, can have no concrete proof or disproof, have determinate truth values? Recourse to ‘in principle possibility’ and idealized beings is rejected as no more acceptable in this domain than appeal to supernatural beings is in biology. A different model of idealization is needed.Less

Avoiding Strict Finitism

Neil Tennant

in The Taming of the True

This chapter shows how to find an acceptable reading of knowability-in-principle in order to avoid visiting upon oneself and one's opposition the overly restrictive consequence that only the ... More

This chapter shows how to find an acceptable reading of knowability-in-principle in order to avoid visiting upon oneself and one's opposition the overly restrictive consequence that only the practically and feasibly knowable can feature in the anti-realist's manifestation requirement on grasp meaning. It discusses the strict finitist's concern about knowability-in-principle, recognitional capacities and compositionality, and feasible verifiability. It also analyses and concept of aspectual recognition and discusses ways of finding the right dispositional conditionals.Less

Quantification and Finitism

Mathieu Marion

in Wittgenstein, Finitism, and the Foundations of Mathematics

This chapter shows that Ludwig Wittgenstein's philosophy of mathematics has much less in common with intuitionism than is usually assumed, although some strategical moves are barely distinguishable. ... More

This chapter shows that Ludwig Wittgenstein's philosophy of mathematics has much less in common with intuitionism than is usually assumed, although some strategical moves are barely distinguishable. At least one such move cannot be overlooked because of its importance for the development of Wittgenstein's philosophy: it concerns quantification theory. This chapter shows that Wittgenstein's stance on quantification was, however, even more radical than that of intuitionists. This is a topic about which Wittgenstein changed his mind around 1929. The search for the source of Wittgenstein's new ideas on quantification leads directly to Hermann Weyl. He actually developed original ideas on topics such as quantification. According to him, propositions containing an existential quantifier ranging over the natural numbers do not possess the full status of judgement.Less

From Truth-Functional Logic to a Logic of Equations

Mathieu Marion

in Wittgenstein, Finitism, and the Foundations of Mathematics

In order to appreciate fully the depth of Ludwig Wittgenstein's commitment to finitism, one ought to look for the causes of his change of mind on quantification. They are to be found in underlying ... More

In order to appreciate fully the depth of Ludwig Wittgenstein's commitment to finitism, one ought to look for the causes of his change of mind on quantification. They are to be found in underlying changes in the notion of analysis which took place in the first steps away from Tractatus Logico-Philosophicus in the manuscripts of 1929. Upon his return to Cambridge in January of that year, Wittgenstein immediately started writing extensively. This was a period of intense intellectual fervor. Early entries in his notebooks indicate that he had started to think afresh some of the problems even before his arrival in Cambridge and that he moved in new directions very quickly. This indicates that Wittgenstein had already given up or was about to give up one of the cental claims of the Tractatus Logico-Philosophicus.Less

Strict Finitism

Mathieu Marion

in Wittgenstein, Finitism, and the Foundations of Mathematics

The philosophical world outside Cambridge discovered the later Ludwig Wittgenstein's philosophy of mathematics with the publication in the 1956 of the Remarks on the Foundations of Mathematics, a ... More

The philosophical world outside Cambridge discovered the later Ludwig Wittgenstein's philosophy of mathematics with the publication in the 1956 of the Remarks on the Foundations of Mathematics, a selection from manuscripts dating from 1938 to 1944. In his 1958 survey paper, Hao Wang promoted strict finitism, which he then called anthropologism, to the rank of a foundational thesis alongside finitism, intuitionism, predicativism, and Platonism, with the later Wittgenstein being enrolled as its most important representative. This has been by far the prevailing view ever since, with some exceptions. A useful preliminary step would be to give a brief characterization of strict finitism. Various strict finitist programmes have been devised in the past, and they do not show much homogeneity: contrary to intuitionism, there is no orthodoxy.Less

Social Constructivism

Ilkka Niiniluoto

in Critical Scientific Realism

Philosophers of science should take seriously the fact, emphasized already by Peirce and Popper, that scientific knowledge is a product of the scientific community. The ontological and ... More

Philosophers of science should take seriously the fact, emphasized already by Peirce and Popper, that scientific knowledge is a product of the scientific community. The ontological and epistemological views associated with the sociology of science have often been interpreted as forms of relativism and anti‐realism. This chapter examines critically the position of the Edinburgh school (Bloor, Barnes). It is argued that the Strong Programme need not be in conflict with realism, but its radically nominalist doctrine of meaning finitism should be rejected. But Latour's social constructivism, if taken literally as an ontological view, cannot be reconciled with scientific realism.Less

Meaning Finitism

Martin Kusch

in Knowledge by Agreement: The Programme of Communitarian Epistemology

Meaning finitism is a communitarian theory of what it is to master a language. It makes community consensus central in the constitution of normativity. Finitism says that terms have no fixed ... More

Meaning finitism is a communitarian theory of what it is to master a language. It makes community consensus central in the constitution of normativity. Finitism says that terms have no fixed extensions. The only extensions there are the ’pseudo‐extensions’ of arrays of exemplars concerning correct use. The categorization of newly‐encountered entities is guided by these exemplars, as well as by interests.Less

Truth

Martin Kusch

in Knowledge by Agreement: The Programme of Communitarian Epistemology

Applies finitism to ’true’ and ’false’. Applying finitism to truth leads to radical and surprising consequences. If ’true’ does not have a fixed extension, then it is simply incoherent to speak of, ... More

Applies finitism to ’true’ and ’false’. Applying finitism to truth leads to radical and surprising consequences. If ’true’ does not have a fixed extension, then it is simply incoherent to speak of, or tacitly rely on, the idea of ’all true statements’ (or ’all true propositions’, or ’all true beliefs’, or ’all true sentences’). And if this idea is illegitimate, then so are traditional ways of thinking about scientific progress (’we are getting ever closer to the truth’). The finitistic theory of meaning constitutes an important constraint on all theories of truth. The finitistic constraint rules out theories of truth that make use of fixed extensions.Less

Reality

Martin Kusch

in Knowledge by Agreement: The Programme of Communitarian Epistemology

Reviews the contemporary debate between realists (’innocent’, ’external’, ’absolute’, ’metaphysical’) and anti‐realists (’irrealism’, ’internal realism’, ’pluralism’, ’relativistic Kantianism’). ... More

Reviews the contemporary debate between realists (’innocent’, ’external’, ’absolute’, ’metaphysical’) and anti‐realists (’irrealism’, ’internal realism’, ’pluralism’, ’relativistic Kantianism’). Criticises ideas and arguments of both sides of the debate where they were based on mistaken semantic assumptions (of fixed extensions). In particular, the chapter rejects the notion of ’one true description of the world’, talk of ’determinate meanings and truth values within a conceptual scheme’, and the idea of once‐and‐for‐all incompatibilities between conceptual schemes.Less

Conclusions

Alexander R. Pruss

in Infinity, Causation, and Paradox

The upshot of the preceding nine chapters is briefly summarized and discussed, offering a view of causal finitism as a via media between finitism and infinitism.

Review of Gödel's Collected Works, Vols. IV and V

Paolo Mancosu

in The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940

This chapter is an essay review of volumes IV and V of Gödel’s Collected Works. Here the aim is to study the heuristics that led to Gödel’s incompleteness theorems by making use of the correspondence ... More

This chapter is an essay review of volumes IV and V of Gödel’s Collected Works. Here the aim is to study the heuristics that led to Gödel’s incompleteness theorems by making use of the correspondence contained in volumes IV and V and to argue that knowledge of the correspondence deepens our understanding of various aspects of the theorems ranging from the role of “truth” and “provability” to specific details of how Gödel gives an exposition of his result in his groundbreaking 1931 article. The chapter also studies the different reactions to Gödel’s incompleteness theorem by Gödel and Bernays, who thought that Hilbert’s program could be salvaged, and those of von Neumann and Herbrand, who saw the result as showing once and for all that Hilbert’s program could not be carried out.Less

Harvard 1940–1941: Tarski, Carnap, and Quine on a Finitistic Language of Mathematics for Science

Paolo Mancosu

in The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940

Tarski, Carnap and Quine spent the academic year 1940–1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among ... More

Tarski, Carnap and Quine spent the academic year 1940–1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among them during the year. These conversations centred around semantical issues related to the analytic/synthetic distinction and on the project of a finitist/nominalist construction of mathematics and science. Carnap’s Nachlass in Pittsburgh contains a set of detailed notes, amounting to more than 80 typescripted pages, taken by Carnap while these discussions were taking place. This Chapter presents a survey of these notes with special emphasis on Tarski’s rejection of the analytic/synthetic distinction, the passage from typed languages to first-order languages, Tarski’s finitism/nominalism, and the construction of a finitist language for mathematics and science.Less

The Fluid Dynamics of Political Culture

Stephen Welch

in The Theory of Political Culture

Chapter 8 accounts for the fluidity of political culture in terms of its discursive dimension. The approach of discursivism has first to be rejected, for it assumes the invariable success of ... More

Chapter 8 accounts for the fluidity of political culture in terms of its discursive dimension. The approach of discursivism has first to be rejected, for it assumes the invariable success of discursive control. The arguments of Chapter 5 show that articulate discourse has a political reach that must always exceed its grasp. Archer’s dualistic theory of culture is found to construe discourse in an excessively abstract manner. In order to capture the causal impact of discourse the theory of ‘meaning finitism’ developed by David Bloor is deployed. This theory argues that discursive agreement creates, through social construction, new objective facts, to which action must adapt. But because these facts are supported only by mutually referring agreement, they are, like the objective conditions of the economic market, subject to unpredictable and chaotic dynamics. The analytic paradigm of the market is then applied to studies of the dynamics of political culture.Less

Infinity, Paradox, and Mathematics

Alexander R. Pruss

in Infinity, Causation, and Paradox

Thomson’s Lamp gives a simple example of a paradox of infinity. Different ways of resolving a paradox in general are considered, and in particular two resolutions of Thomson’s Lamp are considered. ... More

Thomson’s Lamp gives a simple example of a paradox of infinity. Different ways of resolving a paradox in general are considered, and in particular two resolutions of Thomson’s Lamp are considered. According to finitism, there can only be finitely many things, and according to causal finitism, there can only be finitely many causes impinging on an effect. It is argued that finitism fits poorly with mathematics, requires a controversial theory of time to resolve paradoxes, and although it does resolve some additional paradoxes like Hilbert’s Hotel, it is nonetheless not preferable to causal finitism. The additional paradoxes are ones we should just live with, a part of the strangeness of infinity. Additionally, some technical background is given for the book, and a philosophy of mathematics application is given.Less

The Point of Constructivity

Jan von Plato

in The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age

This chapter argues that the best-known statement of finitism is ascribed to Leopold Kronecker (1823–1891). The source of the statement is not Kronecker himself, at least not directly, but rather his ... More

This chapter argues that the best-known statement of finitism is ascribed to Leopold Kronecker (1823–1891). The source of the statement is not Kronecker himself, at least not directly, but rather his editor, Kurt Hensel. Kronecker's finitistic dictum was the basis for the philosophy of mathematics of Thoralf Skolem (1887–1963), a solitary combinatorial genius who started to work with Ernst Schröder's algebra of logic. In his work, he established many of the basic results of lattice theory, such as the independence of the axioms. He also defined what later came to be called Heyting algebras, lattices with an “arrow” operation that imitates implication, and established many basic properties.Less

The Finitist Accountant

David Hatherly, David Leung, and Donald MacKenzie

in Living in a Material World: Economic Sociology Meets Science and Technology Studies

This chapter presents the philosophical roots of finitism, financial reporting in finitist perspective, accounting for economic reality, and rule-governed accounting. Finitism helps people in ... More

This chapter presents the philosophical roots of finitism, financial reporting in finitist perspective, accounting for economic reality, and rule-governed accounting. Finitism helps people in classifying activities, items, and states of all kinds, whereas financial statements include an income statement, a statement of financial position, and a cash flow statement. According to the authors, corporate financial reporting is a major gap in sociological understandings of contemporary economic processes. Financial reporting directly affects the economic health of corporations and economic reality has been considered as an important part of it. The authors have also explained accruals-based current financial reporting, the valuation of a company’s assets, and several approaches for the detection of earnings management.Less

Creative Reasoning

John Turri

in Ad Infinitum: New Essays on Epistemological Infinitism

This chapter defends the unpopular view that inference can create justification. The chapter calls this view inferential creationism. Inferential creationism has been favored by infinitists, who ... More

This chapter defends the unpopular view that inference can create justification. The chapter calls this view inferential creationism. Inferential creationism has been favored by infinitists, who think that it supports infinitism. But it doesn’t. Finitists can and should accept creationism.Less